Abstract

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms Tn:An→Bn for n∈Z0+ and T:⋃j∈Z0+A0j→⋃j∈Z0+B0j, or T:A→(⋃Bn), subject to T(A0n)⊆B0n and Tn(An)⊆Bn, such that Tn converges uniformly to T, and the distances Dn=d(An,Bn) are iteration-dependent, where A0n, An, B0n and Bn are non-empty subsets of X, for n∈Z0+, where (X,d) is a metric space, provided that the set-theoretic limit of the sequences of closed sets {An} and {Bn} exist as n→∞ and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.

Keywords

1 Introduction

The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive and weak contractive mappings and also in related problems of proximal contractions, weak proximal contractions and approximation results and methods [1–15]. The application of the theory of fixed points in stability issues has been proved to be a very useful tool. See, for instance, [16–18] and references therein. This paper is devoted to formulating and proving some further results for more general classes of proximal contractions. The problem of proximal contractions associated with uniformly converging non-self-mappings {Tn}⇉{T} of the form Tn:An→Bn; ∀n∈Z0+, where An and Bn are in general distinct, with a set-theoretic limit of the form T:⋃j∈Z0+A0j→⋃j∈Z0+B0j, provided that the set-theoretic limits of the involved set exist and that the infinite unions of the involved closed sets are also closed. Further related results are obtained for generalized weak proximal and proximal contractions in metric spaces [2, 3, 19], which are subject to certain parametrical constraints on the contractive conditions. Such constraints guarantee that the implying condition of the proximal contraction holds for the proximal sequences so that it can be removed from the analysis [1–3]. Some related generalizations are also given for non-self-mappings of the form T:A→(⋃Bn), subject to a set distance Dn=d(A,Bn), where A and Bn are non-empty and closed subsets of a metric space (X,d) for n∈Z0+, provided that the set-theoretic limit of the sequence of sets {Bn} exists as n→∞. The properties of convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated for the different constraints and the given extension. Application examples are given related to the exact and approximate solutions of compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems [8–11].

1.1 Notation

{Tn}⇉{T} denotes uniform convergence to a limit T of the sequence {Tn} of, in general, non-self-mappings Tn from A to B; ∀n∈Z0+.

Z0+ and Z+ are, respectively, the sets of non-negative and positive integer numbers and R0+ and R+ are, respectively, the sets of non-negative and positive real numbers.

The notation {xn} stands for a sequence with n running on Z0+ simplifying the more involved usual notation {xn}n∈Z0+. A subsequence for indexing subscripts larger than (respectively, larger than or equal to) n0 is denoted as {xn}n>n0 (respectively, as {xn}n≥n0).

Let us establish two definitions of usefulness for the main results of this section.

Definition 2.1 Let (A,B) be a pair of non-empty subsets of a metric space (X,d). A mapping T:A→B is said to be:

(1)

A proximal contraction if there exists a non-negative real number α<1 such that, for all u1,u2,x1,x2∈A, one has

d(ui,Txi)=D(i=1,2)⇒d(u1,u2)≤αd(x1,x2),

where D=d(A,B)=infx∈A,y∈Bd(x,y).

(2)

An asymptotic proximal contraction if there exists a sequence of non-negative real numbers {αn}, with αn<1; ∀n∈Z0+, with αn→α (∈[0,1)) as n→∞ such that, for all sequences {u1n},{u2n},{x1n},{x2n}⊂A,

d(un+1,Txn)=D(n∈Z0+)⇒d(un+2,un+1)≤αnd(xn+1,xn);∀n∈Z0+.

If x,u∈A satisfy d(u,Tx)=D then u∈A and Tx∈B are a pair of best proximity points of T in A and B, respectively. Note that if T:A→B is an asymptotic proximal contraction and the sequence {xn}⊂A is such that (xn+1,Txn)∈A×B; ∀n∈Z0+ is a best proximity pair, then there is a subsequence {xn}n≥n0 of {xn} such that the relation d(xn+2,xn+1)≤αnd(xn+1,xn)≤α¯d(xn+1,xn); ∀n≥n0, holds for some real constant α¯∈[α,1).

Some asymptotic properties of the distances between the sequences of domains and images of the sequences of non-self-mappings Tn:An→Bn; ∀n∈Z0+, which converge uniformly to a limit non-self-mapping T:⋃j∈Z0+A0j→⋃j∈Z0+B0j, are given and proved related to the distance between the domain and image of the limit non-self-mapping.

Lemma 2.2Let(X,d)be a metric space endowed with a homogeneous translation-invariant metricd:X×X→R0+. Consider also a proximal non-self-mapping and a sequence of proximal non-self-mappingsTand{Tn}defined, respectively, byT:⋃j∈Z0+A0j→⋃j∈Z0+B0j, having non-empty images of its restrictionsT:⋃j∈Z0+A0j|A0n→⋃j∈Z0+B0j|B0n; ∀n∈Z0+, andTn:An→Bn; ∀n∈Z0+, whereAn(≠∅)⊆A0nandA0nare non-empty subsets ofXsubject toT(A0n)⊆B0nandTn(An)⊆Bn; ∀n∈Z0+such that the sets of best proximity points:

A0n0={x∈A0n:d(x,y)=Dn0 for some y∈B0n},B0n0={y∈B0n:d(x,y)=Dn0 for some x∈A0n},An0={x∈An:d(x,y)=Dn for some y∈Bn},Bn0={y∈Bn:d(x,y)=Dn for some x∈An}

are non-empty, whereDn0=d(A0n,B0n)andDn=d(An,Bn); ∀n∈Z0+. Let{xn}and{yn}be proximal sequences built in such a way thatx0∈A01, y0∈A1, d(xn+1,Txn)=Dn0andd(yn+1,Tnyn)=Dn; ∀n∈Z0+and define also the error sequence{x˜n}byx˜n=yn−xn; ∀n∈Z0+. Then the following properties hold:

Proof Note that, since d:X×X→R0+ is a homogeneous translation-invariant metric, x0∈A01, y0∈A1, d(xn+1,Txn)=Dn0 and d(yn+1,Tnyn)=Dn; ∀n∈Z0+, one has via induction by using the constraints that xn∈A0n0 (⊂A0n), yn∈An0 (⊂An); ∀n∈Z+ according to

If, furthermore, {xn−yn}→0 and T:⋃j∈Z0+A0j→⋃j∈Z0+B0j is uniformly Lipschitzian in its definition domain then there is a positive real constant kT such that d(Txn,Tyn)≤kTd(xn,yn); and then property (iv) follows from property (iii) since

It turns out that Lemma 2.2 is extendable to the condition {g¯n}→g¯ with the replacements gn→g¯n and g→g¯. Some results on boundedness of distances from points of the domains and their images of Tn:An→Bn; ∀n∈Z0+, and T:⋃j∈Z0+A0j→⋃j∈Z0+B0j and their asymptotic closeness to the set distance are given in the subsequent result.

Lemma 2.3Let(X,d)be a metric space. Consider two sequences{xn}and{yn}built, respectively, under the proximal non-self-mapping{T}and under the sequence of proximal non-self-mappings{Tn}of Lemma 2.2 and assume that{Tn}⇉{T}. Then, for any givenε∈R+, there isN=N(ε)∈Z0+such that the following properties hold:

(i)

d(xn+1,Tnxn)<Dn0+ε, d(yn+1,Tyn)<Dn+ε; ∀n(∈Z0+)>N.

(ii)

If, furthermore,

d(Tyn,Txn)≤K0maxn−mn≤j≤nd(yj,xj)+M;∀n∈Z0+

(2.11)

for some non-negative sequences of integers{mn}withmn≤n; ∀n∈Z0+and some real constantsK0∈[0,1)andM∈R0+, then

whereD¯=D¯(N)=maxN<j<∞(Dj0+Dj), andM′=M′(N)=max1≤j≤Nd(xj,yj)for any given arbitrary finiteN∈Z+.

Proof Assume that the first assertion fails. Then ∃ε∈R+ such that d(xn+1,Tnxn)≥Dn0+ε; ∀n∈Z0+. As a result, since d(xn+1,Txn)=Dn0 by construction and {Tn}⇉{T}, one gets

Dn0+ε≤d(xn+1,Tnxn)≤d(xn+1,Txn)+d(Txn,Tnxn)=Dn0+d(Txn,Tnxn)<Dn0+ε/2;

(2.12)

∀n(∈Z0+)>N and some N∈Z0+, a contradiction. Then the first assertion is true.

Now, assume that the second assertion fails. Then ∃ε∈R+ such that d(yn+1,Tyn)≥Dn+ε; ∀n∈Z0+. As a result, since d(yn+1,Tnyn)=Dn by construction and {Tn}⇉{T}, one gets

Dn+ε≤d(yn+1,Tyn)≤d(yn+1,Tnyn)+d(Tnyn,Tyn)=Dn+d(Tyn,Tnyn)<Dn+ε/2;

(2.13)

∀n(∈Z0+)>N and some N∈Z0+, a contradiction. Then the second assertion is also true and property (i) has been proved. To prove (ii) note that, since {Tn}⇉{T}, for any given ε∈R+, there are Ni=Ni(ε)∈Z0+ for i=1,2 such that

for some non-negative sequences of integers {mn} with mn≤n; ∀n∈Z0+, where d(Txn,Tnxn)<ε/3; ∀n(∈Z0+)>N1, d(Tyn,Tnyn)<ε/3; ∀n(∈Z0+)>N2, N=max(N1,N2), D¯=D¯(N)=maxN<j<∞(Dj0+Dj), and M′ is a non-negative real constant, which is not dependent on n, defined by M′=M′(N)=max1≤j≤Nd(xj,yj). Since K0∈[0,1), one gets

By interchanging the positions of d(xn+1,yn+1) and d(Txn,Tnyn) in the triangle inequality of (2.18), it follows that

|d(Txn,Tnyn)−d(yn+1,xn+1)|≤Dn0+Dn;∀n∈Z0+,

(2.20)

so that if either ⋃n=1∞A0n=⋃n=1∞(An∪A0n), since An⊆A0n; ∀n∈Z0+ is bounded, or ⋃n=1∞(Bn∪B0n) is bounded, both sequences {d(Txn,Tnyn)} and {d(xn,yn)} are bounded. On the other hand, note that the non-negative sequence of integers {mn} might imply the use of an infinite memory in the upper-bounding term of (2.11) if mn≡a∈Z0+; ∀n∈Z0+ or a finite memory of such a bound if m−mn≤m¯; ∀n∈Z0+.

Let (A,B) be a pair of non-empty subsets of a complete metric space (X,d). The set A is said to be approximatively compact with respect to the set B if every sequence {xn}⊂A such that d(y,xn)→d(y,A) for some y∈B has a convergent subsequence.

Theorem 2.5Let a proximal contraction and a sequence of proximal mappings{T}and{Tn}be defined, respectively, byT:⋃n∈Z0+A0n→⋃n∈Z0+B0nhaving non-empty images of its restrictionsT:⋃j∈Z0+A0j|A0n→⋃j∈Z0+B0j|B0n; ∀n∈Z0+andTn:An→Bn; ∀n∈Z0+, whereAn(≠∅)⊆A0nandAnare subsets ofX, where(X,d)is a metric space, subject toT(A0n)⊆B0nandTn(An)⊆Bn; ∀n∈Z0+and the set-theoretic limitslimn→∞An, limn→∞A0n, limn→∞Bnandlimn→∞B0nof the sequences of the sets{An}, {A0n}, {Bn}, {B0n}, respectively, exist and are non-empty being defined in the usual way for any sequence{Zn}of subsetsZn⊆Xas:

limn→∞Zn={z∈X:(zn=1 if z∈Zn)∧(zn=0 if z∉Xn)∧(limn→∞zn=1)}

via the binary set indicator sequences{zn}, satisfy the improper set inclusion conditionlimn→∞An⊆limn→∞A0n. Assume that the sets of best proximity pointsA0n0, B0n0, An0, andBn0are non-empty, whereDn0=d(A0n,B0n)=d(A0n0,B0n0)andDn=d(An,Bn)=d(An0,Bn0); ∀n∈Z0+. Then the following properties hold:

(i)

lim infn→∞T(A0n)∪lim supn→∞T(A0n)⊆limn→∞B0n.

The set-theoretic limits(limn→∞A0n)andT(limn→∞A0n)exist and they are non-empty and closed if the subsetsA0nofXare all closed and⋃n=0∞A0nis closed while it satisfies the following set inclusion constraints:

T(limn→∞A0n)⊆lim supn→∞T(A0n)∩lim supn→∞T(A0n)⊆limn→∞B0n.

If, furthermore, limn→∞T(A0n)exists then it is non-empty and then

T(limn→∞A0n)=limn→∞T(A0n)⊆limn→∞B0n.

The set-theoretic limits(limn→∞An)andT(limn→∞An)exist and they are non-empty and closed if the subsetsAnofX; ∀n∈Z0+are all closed and⋃n=0∞Anis closed, and then

for any sequences{xn}⊂A0n, {yn}⊂Ansatisfyingd(xn+1,Txn)=Dn+10; ∀n∈Z0+andd(yn+1,Tyn)=Dn+1; ∀n(≥n0)∈Z0+for any sequences{xn}⊂A0n, {yn}⊂Anbeing built in such a way thatd(xn+1,Txn)=Dn+10; ∀n∈Z0+andd(yn+1,Tyn)=Dn+1; ∀n(≥n0)∈Z0+.

(iv)

If(X,d)is complete andlimn→∞B0nis approximatively compact with respect tolimn→∞A0nthen there is a convergent subsequence{Txnk}(⊂limn→∞B0n)→Txwherex∈limn→∞A0n0is the limit of{xn}. Iflimn→∞Bnis approximatively compact with respect tolimn→∞Anthen there is a convergent subsequence{Tynk}(⊂limn→∞Bn)→Tywherey∈limn→∞An0is the limit of{yn}. IfD=D0, whereDn0→D0andDn→Dasn→∞, and the above two approximative compactness conditions hold and, furthermore, AnandA0nare closed; ∀n∈Z0+and⋃n=0∞A0nand⋃n=0∞An (it suffices that⋃n=m∞A0nand⋃n=m∞Anbe closed for somem∈Z0+) are closed thenx=y, which is then the unique best proximity point ofTin the limit setlimn→∞An0.

Also, since limn→∞Bn exists, we have lim infn→∞T(An)∪lim supT(An)⊆limn→∞Bn, which follows under a similar reasoning. On the other hand, the existence and non-emptiness of limn→∞A0n, by hypothesis, implies that T(limn→∞A0n) exists since A0n⊂domT; ∀n∈Z0+ so that

if limn→∞T(A0n) exists. If the subsets A0n; ∀n∈Z0+ are all closed and ⋃n=0∞A0n is closed then ⋃n=k∞A0n is closed for any k∈Z+. In order to prove that limn→∞A0n=⋂n=0∞(⋃m=n∞A0m) is closed, that is, an infinite intersection of unions of infinitely many closed sets is closed, it is first proved that all those unions are closed under the given assumption that ⋃n=0∞A0n is closed. Assume this not to be the case, so that there is k∈Z+ such that ⋃n=0∞A0n=(⋃n=0k−1A0n)∪(⋃n=k∞A0n) is closed with ⋃n=k∞A0n not being closed and such that ⋃n=0kA0n is closed, since it is the union of a finite number of closed sets. Then either ⋃n=k−1∞A0n=A0,k−1∪(⋃n=k∞A0n) is not closed or it is closed. In the second case, ⋃n=k∞A0n⊆A0,k−1 since if ⋃n=k∞A0n⊃A0,k−1 is not closed then ⋃n=k−1∞A0n cannot be closed by construction. But, if ⋃n=k∞A0n⊆A0,k−1 then ⋃n=k∞A0n is closed, which contradicts that it is not closed, since A0,k−1 is closed for any k∈Z+. As a result ⋃n=k∞A0n being no closed for any k∈Z+ implies that ⋃n=k−1∞A0n is not closed. By complete induction, it follows that ⋃n=j∞A0n is not closed for any j(∈Z0+)≤k. Thus, ⋃n=0∞A0n is not closed what contradicts that it is closed. As a result, since ⋃n=0∞A0n is non-empty and closed, any infinite union of non-empty closed sets ⋃n=k∞A0n is also non-empty and closed for any k∈Z0+ since ⋃n=0∞A0n is assumed to be closed by hypothesis and it is trivially non-empty. Then limn→∞A0n=⋂n=0∞(⋃m=n∞A0m) is non-empty and closed since it is the infinite intersection of infinitely many unions of non-empty closed sets (but already proved to be non-empty and closed) and there exists the set limit T(limn→∞A0n)=T(⋂n=0∞(⋃m=n∞A0m)) which is now proved to be non-empty. Proceed by contradiction by assuming that it is empty. Then there is some x∈limn→∞A0n such that x∉domT. But then, since x∈limn→∞A0n, limn→∞A0n=⋂n=0∞(⋃m=n∞A0n) is non-empty and closed and all the sets in the sequence {A0n} are closed we have x∈A0n for some n∈Z0+ so that x∈domT from the definition of the non-self-mapping T which contradicts the existence of x∈limn→∞A0n such that x∉domT. It has been proved that limn→∞A0n is closed and T(limn→∞A0n) exists and it is non-empty and closed. The proof that T(limn→∞An)⊆limn→∞Bn is similar to the above one. Now, one proves by contradiction that T(limn→∞An) is non-empty if limn→∞An is non-empty. The limit set limn→∞An is non-empty since the subsets An of X; ∀n∈Z0+ are all closed and ⋃n=0∞An is closed under similar arguments that those used above to prove those properties for limn→∞A0n. Assume that T(limn→∞An) is empty. Since limn→∞An is non-empty, there is x∈limn→∞An such that x∉domT. Since the sets in the sequence {An} are closed, x∈An and x∈A0n, since An⊆A0n, for some n∈Z0+. Thus, x∈domT, from the definition of T and Tx∈T(A0n)∩T(An)⊆B0n∩Bn since T(An)⊆T(A0n)∩Bn⊆B0n∩Bn⊆B0n. Then T(limn→∞An) is non-empty since limn→∞An is non-empty. On the other hand, since limn→∞An⊆limn→∞A0n we have

In the same way, it follows that T(limn→∞An) exists and if, in addition, limn→∞T(An) exists then

limn→∞T(An)=T(limn→∞An)⊆limn→∞Bn.

Hence, property (i) has been proved. Now, build sequences {xn} and {xn′}, and {yn} and {yn′} in X sequences built in such a way that x0,x0′∈A01, y0,y0′∈A1, d(xn+1,Txn)=Dn0, d(xn+1′,Txn′)=Dn+10d(yn+1,Tnyn)=Dn+1 and d(yn+1′,Tyn′)=Dn; ∀n∈Z0+. It follows inductively by using those distance constraints that xn,xn′∈A0n0 (⊂A0n), yn,yn′∈An0 (⊂An); ∀n∈Z+. Note that limn→∞An⊆limn→∞A0n implies the following inclusion of limit sets limn→∞An0⊆limn→∞A0n0 of the sets of best proximity points and the four above limit sets are trivially non-empty. Since T:⋃j∈Z0+A0j→⋃j∈Z0+B0j is a proximal contraction:

(d(xn+1,Txn)=Dn+10;∀n∈Z0+)⇒(d(xn+1,xn+2)≤Kd(xn,xn+1);∀n∈Z0+)

(2.24)

for {xn}⊂A0n0 and some real constant K∈[0,1) and Dn0→D as n→∞, since the limit set limn→∞A0n exists. On the other hand, if {Tn} is a sequence of asymptotic proximal contractions Tn:An→Bn; ∀n∈Z0+ then there is a real sequence {Kn} with a subsequence {Kn}n≥n0⊂[0,K′)⊂[0,1) such that

(d(yn+1,Tnyn)=Dn+1;∀n(≥n0)∈Z0+)⇒(d(yn+1,yn+2)≤K′d(yn,yn+1);∀n(≥n0))

(2.25)

for {yn}n≥n0⊂A0n0 and some n0∈Z+ and some real constant K∈[0,1). Recall the following properties for logical assertions then to be used related to (2.24). Consider the logical propositions Pi; i=1,2. Then

(P2⇒P1)⇔(P1∨¬P2),

(2.26a)

(¬(P2⇒P1))⇔(¬P1∧P2).

(2.26b)

The condition (2.25) corresponds to (2.26a) with P2≡(d(yn+1,Tyn)=Dn+1;∀n∈Z+) and P1≡(d(yn+1,yn+2)≤K′d(yn+1,yn+2);∀n∈Z+). It is now proved by contradiction that {Tn} is a sequence of asymptotic proximal contractions, that is, by assuming that (2.25) is false so that its logical negation

[(d(yn+1,Tnyn)>Dn+1)∨(d(yn+1,yn+2)≤K′d(yn,yn+1));∀n(≥n0)∈Z+]

(2.27)

for {yn}n≥n0⊂A0n0 and some n0∈Z+, obtained from (2.26a)-(2.26b) versus to (2.26a), is true. Then it follows from (2.24) that for any given arbitrary ε∈R+, there are n0,n1=n1(ε)∈Z0+ such that d(Tyn+1,Tnyn+2)<ε; ∀n≥n1 (since {Tn}⇉{T}) and (2.27) hold. Then

Since ε∈R+ is arbitrary, K′∈[0,1), and Dn→D as n→∞ since the limit set limn→∞An exists and D≥D0, since limn→∞An⊆limn→∞A0n, one concludes from (2.28) that

limn→∞d(yn+1,yn+2)=0,{yn}→y(∈limn→∞An),lim infn→∞d(Tyn+1,Tyn)>0.

(2.29)

Since T:⋃n∈Z0+A0n→⋃n∈Z0+B0n and limn→∞An⊆limn→∞A0n, so that y∈limn→∞A0n, since ⋂n≥mA0n is closed by hypothesis for some m∈Z0+ and limn→∞A0n is also closed. Then limn→∞A0n⊂domT so that the possibility of Ty being undefined is excluded and then Ty is defined provided that there is no finite or infinite jump discontinuities at y. If T is continuous at y, then (2.29) leads to the contradiction d(Ty,Ty)>0. Otherwise, if T has a (finite or infinity) jump discontinuity at y, with left and right limits (Ty)− and (Ty)+ (≠(Ty)−), then min(d((Ty)−,(Ty)−),d((Ty)+,(Ty)+))>0, again a contradiction is got. Thus, (2.27) is false so that its negation (2.22) is true. Then {Tn} is a sequence of asymptotic proximal contractions which converge uniformly to the proximal contraction T. Hence, property (ii) has been proved.

Property (iii) is proved by taking into account also (2.24) and the constraints limn→∞A0n0⊆limn→∞A0n and limn→∞An0⊆limn→∞An⊆limn→∞A0n together with the fact that, since ⋂n≥mA0n is closed, limn→∞A0n is closed. Note that the conditions of A0n, An, n∈Z0+ being closed and ⋃n=0∞A0n and ⋃n=0∞An being closed can be relaxed in property (i) to closeness of the sets A0n, An, n(≥m)∈Z0+ and ⋃n=m∞A0n and ⋃n=m∞An for some m∈Z0+ being closed while keeping the corresponding result. Assume this not to be the case. Now, take a subsequence {A0nk} of (non-empty closed) subsets of X such that there is x∈X such that x∈fr(limn→∞A0n), x∉limn→∞A0n, since limn→∞A0n is not closed, and x∈A0nk, the indicator variable of x in any subset in the sequence {A0nk} is x0nk=1 then x∈limk→∞A0nk, since limk→∞x0nk=1, and x∉limn→∞A0n which contradicts limn→∞A0n=limk→∞A0nk. Thus, limn→∞A0n is non-empty and closed. Then one obtains (2.21)-(2.23).

On the other hand, since Dn0→D0 as n→∞ and limn→∞B0n is approximatively compact with respect to limn→∞A0n for some sequence {xn0}⊂limn→∞A0n

d(Txn0,x)→d(limn→∞B0n,x)=D0as n→∞

with x∈limn→∞B0n. From property (i), T(limn→∞An)⊆limn→∞Bn⊆limn→∞B0nT(limn→∞A0n)⊆limn→∞B0n, T(limn→∞An)⊆limn→∞Bn. Then there is a subsequence {Txnk0}(⊂{Txn0})→zx∈cllimn→∞B0n since limn→∞B0n is closed and it is obvious that zx∈cllimn→∞B0n0 (=limn→∞B0n) so that cllimn→∞B0n0≠∅. Assume that limn→∞B0n0 is empty then the best proximity point zx∉limn→∞B0n0, a contradiction so that limn→∞B0n0 is non-empty and it is closed since it is on the boundary of limn→∞B0n0 which is then non-empty and closed. But {xn0}→x (∈limn→∞A0n). Since {xn0} is convergent all its subsequences are convergent to the same limit so that {xnk0}→x. Thus, zx=Tx. Under a similar reasoning, it can be proved under the second given approximative compactness condition that {Tynk0}(⊂{Tyn0})→zy=Ty∈(cllimn→∞Bn) with limn→∞Bn0 being non-empty and closed. If the above both approximative compactness conditions jointly hold and D=D0 then one gets from the contractive condition for the proximal non-self-mapping T that for any given n∈Z0+:

D=d(xn+10,Txn0)=(yn+10,Tyn0)⇒d(xn+10,yn+10)≤Kd(xn0,yn0)

so that {xn0}→x and {yn0}→y implies {d(xn0,yn0)}→d(x,y) so that (1−K)d(x,y)≤0 holds what implies x=y, which has to be necessarily unique from the identity, itself. Hence, property (iv) has been proved. □

Remark 2.6 Note that the conditions of A0n, An, n∈Z0+ being closed and the infinite countable unions ⋃n=0∞A0n and ⋃n=0∞An being closed can be relaxed in Theorem 2.5 (property (i)) to the closeness of the sets A0n, An, n(≥m)∈Z0+ and ⋃n=m∞A0n and ⋃n=m∞An for some m∈Z0+ being closed while keeping the corresponding result.

The following remark is of interest; it concerns a condition of validity of the assumption that a countable union of closed sets is closed used in Theorem 2.5.

Remark 2.7 It can be pointed out that a sufficient condition for the infinite countable union of closed subsets ⋃n=0∞A0n (respectively, ⋃n=0∞An) of a topological space (X,{Bi}) to be closed is that X has the local finiteness property, that is, each point in X has a neighborhood which intersects only finitely many of the closed sets in {A0n} (respectively, in {An}) ([20], pp.29-31). This property can also be applied to a metric space (X,d) since metric spaces are specializations of topological spaces where the metric is used to define the open balls of the topology. More general results guaranteeing that the infinite countable union of closed sets is closed, and equivalently that the infinite countable intersection of open sets is open, stand also for Alexandrov spaces (topological spaces under topologies which are uniquely determined by their specialization preorders) and for P-spaces (the intersection of countably many neighborhoods of each point of the space is also a neighborhood of such a point).

Let us establish two definitions of usefulness for the main results of this section.

Definition 3.1 Let (A,B) be a pair of non-empty subsets of a metric space (X,d). A mapping T:A→B is said to be:

(1)

A generalized asymptotic weak proximal contraction if there are sequences of non-negative real numbers {αn}, with αn∈[0,∞); ∀n∈Z0+ and αn→α (∈[0,1)) as n→∞; and {βn}, with βn∈[0,∞); ∀n∈Z0+ and βn→β (∈[0,∞)) as n→∞ such that, for all sequences {u1},{u2},{x1},{x2}∈A:

A0 and Bn with Dn=d(A,Bn); ∀n∈Z0+ are non-empty and T(A0)⊆B0n; ∀n∈Z0+.

(b)

d(u1n,u2n)≤αnd(x1n,x2n)+βn[d(Tx1n,x2n)−Dn];∀n∈Z0+

(3.2a)

for any sequences {xin}⊂A and {uin}⊂Bn such that d(uin,Txin)=Dn (i=1,2); ∀n∈Z0+ and {αn}, {βn} are real non-negative sequences which satisfy α=lim supn→∞αn<1 and

β=lim supn→∞βn<∞with μ=(α+β)(1+β)+β<1.

(c)

The sequence of set distances {Dn} converges and {Dn−Dn0}→0, where

Dn0≥d(x1n,Tx1n)−(1+αn+βn)d(x1n,x2n);∀n∈Z0+.

(3.2b)

Note if β=0 in Definition 3.1(2), one has the subclass of weak proximal contractions. In this case, one gets by making xn+1=un; ∀n∈Z0+ that d(xn,xn+1)≤αnd(x0,x1); ∀n∈Z0+, so that d(xn,xn+1)→0 as n→∞, since α<1, provided that d(un,Txn)=D and d(xn,Txn)≤(1+α)d(xn,xn+1)+d(un,Txn); ∀n∈Z0+ implying d(xn,Txn)→D as n→∞ and under the conditions that {xn}⊂A and {Txn}⊂B converge they should necessarily converge to best proximity points. Note that Definition 3.1(1) relaxes Definition 3.1(2) and Definition 3.1(3) allows considering weak proximal contractions with sequences built from non-self-mappings which have iteration-dependent image sets.

for any given x0∈A. If μ≤1 and ∑k=0nμn−k(|Dk−Dk0|)<∞ then the sequence {d(xn,xn+1)} is bounded. Thus, property (i) has been proved. The relations (3.6) and (3.7) of property (ii) follow directly from (3.4) of property (i) if μ<1. On the other hand, the triangle inequality and (3.5) lead to (3.11) since

|d(xn+1,Txn)−d(xn,Txn)|≤d(xn,xn+1).

The relation (3.9) follows from (3.7) and (3.8). To prove (3.10), note that if limn→∞(Dn−Dn0)=0, then for any given ε∈R+, there is m=m(ε)∈Z0+ such that supm≤k≤n+m+1|Dk−Dk0|<ε for any n(∈Z+)≥m and then, from (3.10), lim supn→∞d(xn+1,xn)≤βε1−μ. Since ε is arbitrary, the limit limn→∞d(xn,xn+1) exists and limn→∞d(xn,xn+1)=0. This property and (3.8) yield directly limn→∞(|d(xn+1,Txn)−d(xn,Txn)|)=0 and then property (ii) has been fully proved. To prove property (iii), note that (3.11) holds directly from d(A,Bn)=Dn; ∀n∈Z0+ and {xn}⊂A. Also, (3.5) leads to (3.12) by taking into account (3.7). The relation (3.13) follows from (3.7), (3.12), and the relation

d(xn+1,Txn)≤d(xn,Txn)+d(xn+1,xn);∀n∈Z0+.

Hence, property (iii) has been proved. Property (iv) is a direct consequence of property (iii) for the case when d(xn,xn+1)→0 and |Dn−Dn0|→0 as n→∞ including its particular sub-case when {Dn0}→D and {Dn}→D. □

Note that Proposition 3.2 is applicable to the strongly generalized asymptotic weak proximal contraction of Definition 3.1(3) which do not need the fulfilment of the implying part of the logic proposition of Definition 3.1(1)-(2) but the distances of sequences of sets satisfy (3.2b) or, at least, (3.5). The subsequent result is concerned with the existence and uniqueness of best proximity points if, in addition to the assumptions of Proposition 3.2, the set-theoretic limit of the sequence {Bn} exists and is closed and approximatively compact with respect to A.

Theorem 3.3Under all the assumptions of Proposition 3.2 and property (iii), equation (3.10), assume also that(X,d)is complete, thatAandBn; ∀n∈Z0+, are non-empty subsets ofXsuch thatAis closed, A0is non-empty, the set-theoretic limitB:=limn→∞Bnexists, is closed and approximatively compact with respect toA (or the weaker condition thatA0is closed) andT(A0)⊆B0. Assume also that the non-self-mapping restrictionT:A|A1→(⋃Bn)|B, for some subsetA1⊂A, which contains the set of best proximity pointsA0, is a strongly generalized asymptotic weak proximal contraction. ThenT:A|A1→Bhas a unique best proximity point ifμ<1.

Proof Since d(xn+1,xn)→0 as n→∞, {xn}→x from (3.10). Since {xn}⊂A and A is closed we have x∈A. Since Dn≤d(xn+1,Txn)={d(x,Txn)} (→D) since {Dn}→D, because {Bn}→B; and |Dn−d(xn,Txn)|≤d(xn,xn+1); ∀n∈Z0+. Since B is approximatively compact with respect to A and {xn}⊂A, there are y∈A0 and a sequence {x¯n}⊂A, such that {x¯n−xn}→0, then {x¯n}→x since {xn}→x, and {Txn}⊂B such that one gets as n→∞:

d(x¯n,Tx¯n)→d(y,B)=D;d(x¯n,B)→d(y,B)=D;d(xn,B)→d(y,B)=D,

{d(xn,Txn)}→D, {d(xn+1,Txn)}→D and {d(x,Txn)}→D. Since B is approximatively compact with respect to A and {xn}⊂A, the sequence {Txn}⊂⋃Bn, such that Txn∈Bn, has a convergent subsequence {Txnk}→z∈B, since B is also closed and both {Txnk} and {Txnk} have the same limit z∈B. Also, z∈clB0 and d(x,Txnk)→D (=d(x,z)) as k→∞ so that x is a best proximity point of T:A|A1→B. Note that since the limit set B exists, it is by construction the infinite union of intersections of the form B=⋃n=0∞⋂m≥n∞Bm so that B0⊆B⊆⋃Bn so that there is a restriction T:A|A1→(⋃Bn)|B for some A1⊆A which is non-empty. Assume not so that A1=∅. If A1=∅ then A0 is also empty which is impossible then A1≠∅. It is now proved by contradiction that the best proximity point is unique. Assume this not to be the case so that there are two best proximity points x, y such that there are two sequences {xn}→x and {yn}→y contained in A. Since T:A|A1→(⋃Bn)|B is a strongly generalized asymptotic weak proximal contraction, one gets from the implied logic proposition of (3.1) with u=x and v=y and Dn=D that

4 Examples

Two examples are described to the light of proximal contractions. The first one is concerned with the solution of algebraic systems which can have more or less unknown than equations and which can be compatible or not. The second one is referred to an identification problem of a discrete dynamic system whose parameters are unknown and which can be subject to unmodeled dynamics and/or exogenous noise which makes not possible, in general, an exact identification.

Example 4.1 (Moore-Penrose pseudo-inverse)

The problem of solving either exactly or approximately a linear system of algebraic equations is very important and it appears in many engineering and scientific applications. It is possible to focus it to the light of best proximity points of non-self-mappings as follows. Consider the linear algebraic system Cx=e where C∈Rn×p (a real matrix of order n×p) and e∈Rp. It is known from the Rouché-Frobenius theorem from Linear Algebra that a solution x∈Rp exists if rank(C,e)=rankC. The solution is unique given by x=C−1e if p=n, and rankC=p with the algebraic system being determined compatible. If rank(C,e)=rankC=q≤min(n,p)≠n then there are infinitely many solutions and the algebraic system is indetermined compatible being, in particular, overdetermined if n>p and undetermined if n<p. If rank(C,e)>rankC then the algebraic system is incompatible. A more general setting is CX=E, where C∈Rn×p, E∈Rn×q are given and X∈Rp×q is a solution which exists if and only if rankC=rank(C⋮E). The following cases hold:

(a)

If rankC=rank(C⋮E)≠p, then C+CE≠E, and there are infinitely many solutions of the form X=C+E+(I−C+C)W with C+ being the Moore-Penrose pseudo-inverse of C. The domain of the non-self-mapping TC:A→B, represented by the matrix C, can be restricted to A={X=C+E+(I−C+C)W:W∈Rp×q}. To close a proper formalism we extend the matrices in A to matrices A¯⊂Rn×(max(p,q)−q) and those in B to B¯⊂Rn×(max(p,q)−p) by adding zero columns (if p≠q either A≠A¯ or B≠B¯) and we consider them as subsets of X≡Rn×(max(p,q)−q) and consider the metric space (X,d) with d being the Euclidean metric so that D=0 with the sets of best proximity points of A¯ and B¯ being:

A¯0 is the set of solutions of the compatible indeterminate algebraic system.

(b)

If rankC=rank(C⋮E)=p then the solution X=C+E is unique and A¯0=A¯={(C+E⋮0):0∈Rn×(max(p,q)−q)} consist of one element which is the unique solution.

(c)

If min(p,n)≥rankC<rank(C⋮E) then C+CE≠E and the algebraic system is incompatible. By considering (X,d) as a Banach space endowed with the Euclidean norm, we can check for the best solution which minimizes ∥CX−E∥ over X∈Rp×q if it exists. Such a solution exists in a least-squares sense and it is unique if n≥rankC=p<rank(C⋮E), since CTC is non-singular, of order p, and C+=(CTC)−1CT, and Xˆ=(CTC)−1CTE minimizes ∥CX−E∥ over the set of matrices X∈Rp×q so that D=d(A¯,B¯)=∥((CTC)−1CT−I)E∥, the sets of best proximity points of A¯ being

A¯0=A¯={X¯=(X⋮0):X=(CTC)−1CTE,0∈Rn×(max(p,q)−q)},

since (I−C+C)=0 and B¯0 is as the above one of case (a). A¯0 is the best solution of the incompatible algebraic system.

The pseudo-inverse can be calculated without inverting CTC by the iterative process:

Cn+1+=(2I−Cn+C)Cn+;n∈Z+;C0+such that C0+C=(C0+C)∗

(so-called Ben-Israel-Cohen, or hyper-power sequence, iterative method [21]). It follows that Cn+→C+ as n→∞ since (1) C+ is unique; and (2) the iterative process satisfies the pseudo-inverse properties C+=C+CC+ and C+C=(C+C)∗ under the replacement Cn+→C+; ∀n∈Z0+. By using the iterative process, we can also define sequences of sets and associate sequences of distances by:

B¯0 and A¯0n are unique, if the pseudo-inverse C+=(CTC)−1CT exists for the given initial C0+. However, if rankC=rank(C⋮E)≠p, so that E≠Cn+CnE (case (a) - incompatible algebraic system) then if we use the iterative procedure: